Galois closure data for extensions of rings

Abstract

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of ${\mathrm{S}}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $\phi :(A^{\otimes n}{)}^G\to R$ satisfying certain properties, and we associate to a closure datum $\phi$ a closure algebra $A^{\otimes n}{\otimes }_{(A^{\otimes n}{)}^G}R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corresponding Galois group. We describe G-closure data and algebras of finite étale algebras over a general connected ring $R$ in terms of the corresponding finite sets with continuous actions by the étale fundamental group of $R$. We show that if $2$ is invertible, then ${\mathrm{A}}_n$-closure data for free extensions correspond to square roots of the discriminant, and that ${\mathrm{D}}_4$-closure data for quartic monogenic extensions correspond to roots of the cubic resolvent.

arXiv commentThis is a condensed, updated, and revised version of the author's Ph.D.\ thesis